On some medial semigroups with an associate subgroup

Abstract

Let S be a semigroup and s,t∈S. We say that t is an associate of s if s=sts. If S has a maximal subgroup G such that every element s of S has a unique associate in G, say s ^∗, we say that G is an associate subgroup of S and consider the mapping s→s ^∗ as a unary operation on S. In this way, semigroups with an associate subgroup may be identified with unary semigroups satisfying three simple axioms. Among them, only those satisfying the identity (st)^∗=t ^∗ s ^∗, called medial, have a structure theorem, due to Blyth and Martins.

We introduce several relations germane to the presence of this unary operation. Next we characterize medial, orthodox, cryptic and medial, and cryptic and orthodox semigroups with an associate subgroup in terms of a construction and several choices of a basis of their identities. In addition, we characterize medial monoids and discuss F-inverse semigroups in terms of a construction and bases for their systems of identities.

We establish an isomorphism between the lattice of varieties of cryptic medial monoids and the direct product of the lattice of band monoids and the lattice of group varieties. An embedding into idempotent generated semigroups is proved as well.